An ergodic approach to Laplace transforms on time scales (Q2033172)

The authors examine the region of convergence for the Laplace transform on time scales (i.e., closed subsets of the real axis). They employ an (what they call) ergodic approach in order to both calculate (1) the region of convergence and (2) the inversion of the transform. It turns out that the problem (1) is closedly related to the problem of determining regions of stability for the time scale exponential function \(e_z(t,t_0)\). Concerning the problem (2) the authors use ergodic techniques to arrive at a solution, and so in the current paper they expound upon these notions and show how they can be used to think about the Laplace transform in an effective and efficient manner. \par The paper is carefully written and nicely illustrated using various related examples. In particular, its introduction contains an overview of the current state of the time scales theory.